Complex geometry and structured chaos

Fractal geometry has infiltrated popular culture since it was formalized in the early 80’s from the works of Benoit Mandelbrot. While it has been used to study the form of cities by researchers such as Pierre Frankhauser and Michael Batty, the insights to be drawn from this field of mathematics have not yet penetrated the field of urbanism, defined as the construction of cities. Connecting the fractal city by Nikos Salingaros approaches the topic by asking what type of city is fractal, without going into depth as to how a fractal is made. Christopher Alexander, in his second tome of The Nature of Order, The Process of Creating Life, begins to develop profound ideas on the topic, which he had hinted to in The Oregon Experiment and A New Theory of Urban Design.

The basic quality of fractal geometry is that it is recursively-defined geometry; it must be described in terms of itself. A triangle, in basic euclidean geometry, is defined by the connection of three vectors at their extremities. Euclidean geometry is built up by combining basic elements into different shapes. A point becomes a line, which becomes a triangle, which becomes several different kinds of polygons, and so on. (A famous introductory architecture textbook, Architecture: Form, Space and Order by Francis D. K. Ching uses this method.) Fractal geometry does not take this approach of combination. Instead of using a triangle to make a square, in fractal geometry we use a triangle to make another triangle, such as this Sierpinski triangle:

At each step we use the results of the previous step and repeat some procedure, in this case either adding two copies of the previous object below the current one (composition) or replacing the three large triangles each by a copy of the object (decomposition). Both approaches will generate the Sierpinski triangle over an infinite number of repetitions.

The words generate and infinite are very important. It is these two words that make fractal geometry so completely different from euclidean geometry, which can be drawn instantaneously. Because fractal geometry is recursive, it is in theory infinitely complex, and the only way to see what a fractal object will look like is to run the computation that generates it until we grow tired of watching the process unfold. It is, by its own nature, surprising, unpredictable, and thus emergent.

The idea of objects substituting themselves for copies of themselves is nothing that revolutionary. It is the basic process that underlies all living things. In a living system a starting point, the embryo, contains a program, DNA, that will be multiplied into trillions of cells. The cells follow the transformations described by their DNA codes by taking certain actions depending on their environmental factors and previous states. (Alexander uses the example of a bone, whose shape evenly distributes structural stress across its surface, by claiming that the form of a bone emerges from a program telling cells to add bone mass where the stress is most intense.) Because living systems are the result of recursive transformations, it should not be a shock that they exhibit the properties of fractal geometry. The inward-out, decentralized growth of living things makes possible complexity in nature. Benoit Mandelbrot made this obvious when he wrote The Fractal Geometry of Nature, a book that pretty much started the fractal revolution by providing a mathematical framework for understanding real physical space.

Coming back to our preferred subject matter, cities and their construction, there is something very profound going on in the construction of cities if Mr. Frankhauser and Batty can calculate an index of “fractality” that is higher for some cities than for others. It means that in the process of building cities humans have unconsciously created complexity by adopting certain processes at certain times, and forgotten or abolished them at other times. That is a fascinating topic of discussion, a debate which has been at the heart of the profession going back perhaps further than the vastly different paths taken by Haussmann and Cerda in the 19th century, and one that must be at the heart of the profession of urbanism today more than ever. What is the alternative to planning and deliberate design of cities, which have nothing but a history of failure to show for themselves?

I’ve explained in a previous article how the very purpose of building cities is to create networks of buildings that handle chaos, the everyday uncertainty of future needs against the permanence of individual buildings. The very act of growing a city is an act of differentiation, creating something different from what currently exists as part of the city’s network of buildings in order to fulfill a need that the existing building stock cannot fulfill. (In other words, adaptation to changing circumstances.) These differentiations can be the creation of new public spaces, such as the boulevard construction initiated by Haussmann that provided large-scale connectivity to the city of Paris, as well as the construction of new, unforeseen buildings. But this is where the architectural design approach to urbanism runs into a major problem: how can beauty and order be created out of something that must necessarily be different from everything else?

The answer to that question is hidden in Benoit Mandelbrot’s greatest discovery, the Mandelbrot Set.

The algorithm that generates the Mandelbrot Set is, like those behind all the beautiful complex structures, extremely simple. It is a chaotic algorithm that “spins” within the boundaries of 2 and -2. For given coordinates in the plane made up of the normal and complex numbers, each coordinate will either spin forever in the orbit of radius 2, or escape after a determined number of iterations. The coordinates which never escape are defined as being part of the set.

A black-on-white picture of the set is by itself very intriguing, but the true beauty of it is not revealed until we apply a system of transformation to the coordinates that were thrown out of it. If, each time we throw out a pair of coordinates, we assign to it a number equivalent to the number of iterations it took to figure out it didn’t belong in the set, we will form groups of chaotic equivalence. And once we apply a single, shared transformation (a “DNA code” for the chaotic equation) to these sets, in this case defining a color for each iteration that threw out some coordinates, applying this color to these coordinates while drawing the Mandelbrot Set, we will generate this kind of geometry:

This is what I refer to as structured chaos. By applying a shared system of transformation to chaotic events, we obtain complex geometry. Shared transformations are the source of the new symmetric property of fractals known as self-similarity, and they are also the source of the wholeness and beauty of those chaotic systems called life, including cities.

Reflecting on the way cities have been built throughout history, the most beautiful places have been those that have shared transformations while creating differentiations. The city of Venice, which continues to inspire architects despite their inability to live up to its beauty, is a perfect example of shared transformations creating wholeness out of chaos. But to understand how to create symmetry by self-similarity, one has to be able to decompose buildings into their different scales and chaotic fields (differentiated elements).

The tradition of teaching the classical orders in architecture was once an imperfect approach to granting architects this skill. The classical orders are one form of transformation system, where large-scale elements, the column, the entablature, are decomposed into smaller-scale elements, the capital, the shaft, which form the large scale elements. And so when many architects, trained to share this transformation system as part of their skill set, worked on completely different buildings, their work could easily form a larger whole; whenever they hit similar problems, they would employ the similar solution they were trained to employ. While two buildings may have completely different sizes or roofs, or one could have a bell tower while the other didn’t, if both buildings had windows and columns, the windows and columns would be made the same way, and thus symmetrical to each other. This is how every building in a city was tied together in a web of geometric relationships, and it is the density of these relationships that gave cities their quality of wholeness and beauty. This property goes beyond the scale of the classical orders. It is also true of ancient Asian cities, or the mythic New York of the 1940’s.

Sadly the unending race for pure originality and the abandonment of hierarchical geometry by the architectural profession has made the creation of such cityscapes impossible. This has made the modern architectural profession largely parasitic of the city, and their professional ruin easily explained. Given what we now understand about generating geometric wholeness out of chaos, is there anything that justifies, other than a desire for euclidean perfection, the creation of rigid euclidean plans for cities? I have not been able to find any. The work of urbanism must be about two fundamental aspects: defining a system of transformations that will apply to all unforeseeable acts of construction in the city, at all scales, and creating the connective public space that will bind the different buildings together.

Returning to Nikos Salingaros’ question, the kind of a city that is a fractal is the kind that is made by applying shared transformations to chaotic events. This is the holy grail of urbanism in the 21st century. With this knowledge we can finally surpass the classical city, bury the demons of Le Corbusier and the C.I.A.M. while embracing all that technology has to offer to urbanism.